Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B

Average Accuracy: 68.4% → 99.8%

Time: 16.5s

Precision: binary64

Cost: 26697

Math

\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -580000 \lor \neg \left(z \leq 112\right):\\ \;\;\;\;x + \frac{y}{\frac{101.23733352003822}{z \cdot z} + \left(14.431876219268936 + \frac{-15.646356830292042}{z}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, y, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  x
  (/
   (*
    y
    (+
     (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
     0.279195317918525))
   (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -580000.0) (not (<= z 112.0)))
   (+
    x
    (/
     y
     (+
      (/ 101.23733352003822 (* z z))
      (+ 14.431876219268936 (/ -15.646356830292042 z)))))
   (fma
    (/
     (fma z (fma z 0.0692910599291889 0.4917317610505968) 0.279195317918525)
     (fma z (+ z 6.012459259764103) 3.350343815022304))
    y
    x)))

C

double code(double x, double y, double z) {
	return x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304));
}

↓

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -580000.0) || !(z <= 112.0)) {
		tmp = x + (y / ((101.23733352003822 / (z * z)) + (14.431876219268936 + (-15.646356830292042 / z))));
	} else {
		tmp = fma((fma(z, fma(z, 0.0692910599291889, 0.4917317610505968), 0.279195317918525) / fma(z, (z + 6.012459259764103), 3.350343815022304)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304)))
end

↓

function code(x, y, z)
	tmp = 0.0
	if ((z <= -580000.0) || !(z <= 112.0))
		tmp = Float64(x + Float64(y / Float64(Float64(101.23733352003822 / Float64(z * z)) + Float64(14.431876219268936 + Float64(-15.646356830292042 / z)))));
	else
		tmp = fma(Float64(fma(z, fma(z, 0.0692910599291889, 0.4917317610505968), 0.279195317918525) / fma(z, Float64(z + 6.012459259764103), 3.350343815022304)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := If[Or[LessEqual[z, -580000.0], N[Not[LessEqual[z, 112.0]], $MachinePrecision]], N[(x + N[(y / N[(N[(101.23733352003822 / N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(14.431876219268936 + N[(-15.646356830292042 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * N[(z * 0.0692910599291889 + 0.4917317610505968), $MachinePrecision] + 0.279195317918525), $MachinePrecision] / N[(z * N[(z + 6.012459259764103), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]

Target

Original	68.4%
Target	99.4%
Herbie	99.8%

\[\begin{array}{l} \mathbf{if}\;z < -8120153.652456675:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291889\right) \cdot y - \left(\frac{0.40462203869992125 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z < 6.576118972787377 \cdot 10^{+20}:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)\right) \cdot \frac{1}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291889\right) \cdot y - \left(\frac{0.40462203869992125 \cdot y}{z \cdot z} - x\right)\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if z < -5.8e5 or 112 < z

   1. Initial program 37.7%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

   2. Simplified 49.6%

      \[\leadsto \color{blue}{x + \frac{y}{\frac{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}}} \]
      Proof

      [Start] 37.7	\[ x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
      associate-/l* [=>] 49.6	\[ x + \color{blue}{\frac{y}{\frac{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}}} \]
      fma-def [=>] 49.6	\[ x + \frac{y}{\frac{\color{blue}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}}{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}} \]
      fma-def [=>] 49.6	\[ x + \frac{y}{\frac{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}{\color{blue}{\mathsf{fma}\left(z \cdot 0.0692910599291889 + 0.4917317610505968, z, 0.279195317918525\right)}}} \]
      fma-def [=>] 49.6	\[ x + \frac{y}{\frac{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, z, 0.279195317918525\right)}} \]

   3. Taylor expanded in z around inf 99.8%

      \[\leadsto x + \frac{y}{\color{blue}{\left(14.431876219268936 + 101.23733352003822 \cdot \frac{1}{{z}^{2}}\right) - 15.646356830292042 \cdot \frac{1}{z}}} \]

   4. Simplified 99.8%

      \[\leadsto x + \frac{y}{\color{blue}{\frac{101.23733352003822}{z \cdot z} + \left(14.431876219268936 + \frac{-15.646356830292042}{z}\right)}} \]
      Proof

      [Start] 99.8	\[ x + \frac{y}{\left(14.431876219268936 + 101.23733352003822 \cdot \frac{1}{{z}^{2}}\right) - 15.646356830292042 \cdot \frac{1}{z}} \]
      cancel-sign-sub-inv [=>] 99.8	\[ x + \frac{y}{\color{blue}{\left(14.431876219268936 + 101.23733352003822 \cdot \frac{1}{{z}^{2}}\right) + \left(-15.646356830292042\right) \cdot \frac{1}{z}}} \]
      +-commutative [=>] 99.8	\[ x + \frac{y}{\color{blue}{\left(101.23733352003822 \cdot \frac{1}{{z}^{2}} + 14.431876219268936\right)} + \left(-15.646356830292042\right) \cdot \frac{1}{z}} \]
      associate-+l+ [=>] 99.8	\[ x + \frac{y}{\color{blue}{101.23733352003822 \cdot \frac{1}{{z}^{2}} + \left(14.431876219268936 + \left(-15.646356830292042\right) \cdot \frac{1}{z}\right)}} \]
      associate-*r/ [=>] 99.8	\[ x + \frac{y}{\color{blue}{\frac{101.23733352003822 \cdot 1}{{z}^{2}}} + \left(14.431876219268936 + \left(-15.646356830292042\right) \cdot \frac{1}{z}\right)} \]
      metadata-eval [=>] 99.8	\[ x + \frac{y}{\frac{\color{blue}{101.23733352003822}}{{z}^{2}} + \left(14.431876219268936 + \left(-15.646356830292042\right) \cdot \frac{1}{z}\right)} \]
      unpow2 [=>] 99.8	\[ x + \frac{y}{\frac{101.23733352003822}{\color{blue}{z \cdot z}} + \left(14.431876219268936 + \left(-15.646356830292042\right) \cdot \frac{1}{z}\right)} \]
      associate-*r/ [=>] 99.8	\[ x + \frac{y}{\frac{101.23733352003822}{z \cdot z} + \left(14.431876219268936 + \color{blue}{\frac{\left(-15.646356830292042\right) \cdot 1}{z}}\right)} \]
      metadata-eval [=>] 99.8	\[ x + \frac{y}{\frac{101.23733352003822}{z \cdot z} + \left(14.431876219268936 + \frac{\color{blue}{-15.646356830292042} \cdot 1}{z}\right)} \]
      metadata-eval [=>] 99.8	\[ x + \frac{y}{\frac{101.23733352003822}{z \cdot z} + \left(14.431876219268936 + \frac{\color{blue}{-15.646356830292042}}{z}\right)} \]

   if -5.8e5 < z < 112

   1. Initial program 99.6%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

   2. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, y, x\right)} \]
      Proof

      [Start] 99.6	\[ x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
      +-commutative [=>] 99.6	\[ \color{blue}{\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x} \]
      associate-*r/ [<=] 99.8	\[ \color{blue}{y \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} + x \]
      *-commutative [<=] 99.8	\[ \color{blue}{\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \cdot y} + x \]
      fma-def [=>] 99.8	\[ \color{blue}{\mathsf{fma}\left(\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, y, x\right)} \]
      *-commutative [=>] 99.8	\[ \mathsf{fma}\left(\frac{\color{blue}{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right)} + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, y, x\right) \]
      fma-def [=>] 99.8	\[ \mathsf{fma}\left(\frac{\color{blue}{\mathsf{fma}\left(z, z \cdot 0.0692910599291889 + 0.4917317610505968, 0.279195317918525\right)}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, y, x\right) \]
      fma-def [=>] 99.8	\[ \mathsf{fma}\left(\frac{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, y, x\right) \]
      *-commutative [=>] 99.8	\[ \mathsf{fma}\left(\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\color{blue}{z \cdot \left(z + 6.012459259764103\right)} + 3.350343815022304}, y, x\right) \]
      fma-def [=>] 99.8	\[ \mathsf{fma}\left(\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\color{blue}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}}, y, x\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -580000 \lor \neg \left(z \leq 112\right):\\ \;\;\;\;x + \frac{y}{\frac{101.23733352003822}{z \cdot z} + \left(14.431876219268936 + \frac{-15.646356830292042}{z}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, y, x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	20424

\[\begin{array}{l} t_0 := 14.431876219268936 + \frac{-15.646356830292042}{z}\\ \mathbf{if}\;z \leq -8200000000000:\\ \;\;\;\;x + \frac{y}{t_0}\\ \mathbf{elif}\;z \leq 112:\\ \;\;\;\;x + \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right) \cdot \frac{y}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{101.23733352003822}{z \cdot z} + t_0}\\ \end{array} \]

Alternative 2
Accuracy	99.7%
Cost	1609

\[\begin{array}{l} \mathbf{if}\;z \leq -580000 \lor \neg \left(z \leq 112\right):\\ \;\;\;\;x + \frac{y}{\frac{101.23733352003822}{z \cdot z} + \left(14.431876219268936 + \frac{-15.646356830292042}{z}\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(0.279195317918525 + z \cdot \left(0.4917317610505968 + z \cdot 0.0692910599291889\right)\right)}{3.350343815022304 + z \cdot \left(z + 6.012459259764103\right)}\\ \end{array} \]

Alternative 3
Accuracy	99.4%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;z \leq -5.4 \lor \neg \left(z \leq 5.2\right):\\ \;\;\;\;x + \frac{y}{14.431876219268936 + \frac{-15.646356830292042}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\left(z \cdot 0.39999999996247915 + 12.000000000000014\right) + \left(z \cdot z\right) \cdot -0.10095235035524991}\\ \end{array} \]

Alternative 4
Accuracy	99.5%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;z \leq -5.4 \lor \neg \left(z \leq 5.3\right):\\ \;\;\;\;x + \frac{y}{\frac{101.23733352003822}{z \cdot z} + \left(14.431876219268936 + \frac{-15.646356830292042}{z}\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\left(z \cdot 0.39999999996247915 + 12.000000000000014\right) + \left(z \cdot z\right) \cdot -0.10095235035524991}\\ \end{array} \]

Alternative 5
Accuracy	60.5%
Cost	1116

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-111}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-201}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-283}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-170}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-126}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-107}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-26}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Accuracy	99.1%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -5.4 \lor \neg \left(z \leq 6.2\right):\\ \;\;\;\;x + \frac{y}{14.431876219268936}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z \cdot 0.39999999996247915 + 12.000000000000014}\\ \end{array} \]

Alternative 7
Accuracy	99.3%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -5.4 \lor \neg \left(z \leq 6.4\right):\\ \;\;\;\;x + \frac{y}{14.431876219268936 + \frac{-15.646356830292042}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z \cdot 0.39999999996247915 + 12.000000000000014}\\ \end{array} \]

Alternative 8
Accuracy	60.1%
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-111}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-144}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-108}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-19}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Accuracy	78.5%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-286} \lor \neg \left(x \leq 4.9 \cdot 10^{-284}\right):\\ \;\;\;\;x + \frac{y}{12.000000000000014}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \end{array} \]

Alternative 10
Accuracy	98.8%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -5.4 \lor \neg \left(z \leq 5.6\right):\\ \;\;\;\;x + \frac{y}{14.431876219268936}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{12.000000000000014}\\ \end{array} \]

Alternative 11
Accuracy	50.3%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023140 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< z -8120153.652456675) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x)) (if (< z 6.576118972787377e+20) (+ x (* (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (/ 1.0 (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x))))

  (+ x (/ (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))